MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1cocnv1 Structured version   Unicode version

Theorem f1cocnv1 5734
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv1  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)

Proof of Theorem f1cocnv1
StepHypRef Expression
1 f1f1orn 5714 . 2  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ococnv1 5733 . 2  |-  ( F : A -1-1-onto-> ran  F  ->  ( `' F  o.  F
)  =  (  _I  |`  A ) )
31, 2syl 16 1  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    _I cid 4522   `'ccnv 4906   ran crn 4908    |` cres 4909    o. ccom 4911   -1-1->wf1 5480   -1-1-onto->wf1o 5482
This theorem is referenced by:  f1eqcocnv  6057  domss2  7295  diophrw  26855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490
  Copyright terms: Public domain W3C validator